Question: What is the remainder when $5^{137}$ is divided by 8?
Start with small exponents and look for a pattern.  We have $5^1\equiv 5\pmod{8}$ and $5^2\equiv 1\pmod{8}$.  We can multiply both sides of $5^2\equiv 1\pmod{8}$ by 5 to find that $5^3\equiv 5\pmod{8}$.  Multiplying both sides by 5 again, we find $5^4\equiv 1\pmod{8}$.  We see that every odd power of 5 is congruent to 5 modulo 8, and every even power is congruent to 1 modulo 8.  Therefore, $5^{137}$ leaves a remainder of $\boxed{5}$ when divided by 8.